McCombs Math 381
Proofs and Set Operations
Given finite sets
A
and
B
.
1.
A
!
B
=
x

x
"
A
or
x
"
B
{ }
2.
A
!
B
=
x

x
"
A
and
x
"
B
{ }
3.
A
!
B
=
x
,
y
( )

x
"
A
,
y
"
B
{ }
4.
A
!
B
=
x

x
"
A
and
x
#
B
{ }
5.
A
!
B
=
A
"
B
( )
#
B
"
A
( )
6.
P A
( )
=
S

S
!
A
{ }
7.
A
=
x

x
!
A
{ }
8.
P A
( )
=
2
A
9.
A
!
B
=
A
+
B
"
A
#
B
10.
A
!
B
=
A
"
B
11.
A
!
B
"
C
( )
=
A
!
B
( )
#
A
!
C
( )
12.
A
!
B
"
C
( )
=
A
!
B
( )
#
A
!
C
( )
Proof Strategies
1.
To prove
A
!
B
, we show
x
!
A
"
x
!
B
.
2.
To prove
A
=
B
, we show
A
!
B
and
B
!
A
.
Examples:
Prove each statement.
1.
Given set
A
,
!"
A
.
Proof:
We need to show that
x
!"#
x
!
A
.
This statement is vacuously true since
x
!"
is a FALSE statement.
2.
There is only one empty set.
Proof:
Suppose we have two sets
E
1
and
E
2
each of which is empty.
By Example 1 above,
E
1
empty implies
E
1
!
E
2
.
Similarly,
E
2
empty implies
E
2
!
E
1
.
Thus,
E
1
=
E
2
. Therefore, there is only one empty set.
3.
Given integers
c
and
d
, let
C
=
x
!
Z
:
x
c
{ }
, and
D
=
x
!
Z
:
x
d
{ }
.
Prove that
C
!
D
if and only if
c
d
.
Proof:
Part 1:
C
!
D
"
c
d
.
Assume
C
!
D
. This means that every element in
C
is also an
element in
D
. Note that the integer
c
!
C
since
c
c
.
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 Summer '11
 NA
 Sets, Rational number, positive rational number

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